Observing metal-insulator quantum criticality in multilayer MoS2: Soft Coulomb gap .... shows the optical and AFM images, and thickness profile of this sample.
Observing metal-insulator quantum criticality in multilayer MoS2: Soft Coulomb gap and asymmetric scaling
Moon et al.
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Supplementary Information Observing metal-insulator quantum criticality in multilayer MoS2: Soft Coulomb gap and asymmetric scaling Byoung Hee Moon1,2β *, Jung Jun Bae 1β , Min-Kyu Joo1, Homin Choi1,2, Gang Hee Han1, Hanjo Lim3, Young Hee Lee1,2*
1
Center for Integrated Nanostructure Physics, Institute for Basic Science (IBS), Suwon 16419,
Republic of Korea 2
Department of Energy Science, Sungkyunkwan University, Suwon 16419, Republic of Korea.
3
Institute for Basic Science (IBS), Daejeon 34047, Republic of Korea
Supplementary Note 1. Determination of carrier density In order to check how reliable the βsimple approximationβ is, we made another multilayer MoS2 device of similar thickness, ~ 4 nm as shown in the inset of Supplementary Fig. 1 below. Supplementary Fig. 1a shows the current response to the backgate bias VBG. The threshold voltage Vth was determined as usual by extrapolating the linear part as in this figure, giving Vth ~11 V. For comparison, we performed Hall measurements. At each fixed magnetic field B from 0 to 8 T by 0.5 T step, we did backgate bias sweeps and measured Hall voltages. From the slope in VHall vs. B plot for each VBG, we show an example for chosen backgate bias in Supplementary Fig. 1b and the carrier density n2D for each VBG was calculated as shown in Supplementary Fig. 1c. n2D values are rather scattered but their VBG dependences show the decent linear behavior for both Vds = 0.5 and 1 V. The solid line in this figure is from the approximation πΏπΏ2D = πΆox (πBG β πth )βπ using Vth = 11 V and Cox = 11.5Γ10-8 F/cm2 for 300 nm SiO2. The
effective oxide capacitance from the Hall measurement is nearly identical to the geometric value while the carrier density obtained from geometry undereatimates the value of n2D. However, we note that the determination of the critical exponent values is insensitive to the constant offset of 2
n2D due to Ξ΄πΏπΏ β‘ (πΏπΏ2D β πΏπΏc )βπΏπΏc .
a
b
3
6 VBG 16 24 32 40
5
VHall (mV)
Ids (Β΅A)
2
1
4 3 2 1
Vth ~ 11 V
0
0 -10 0
10 20 30 40
VBG (V)
c
0
2
4
6
8
B (T)
d 30
Vds = 0.5 V Vds = 1.0 V
Β΅Hall (cm2V-1s-1)
n2D (1012 cm-2)
3 2 1
20
10 Vds = 0.5 V Vds = 1.0 V
0 0
10
20
VBG (V)
30
40
0
0
10
20
30
40
VBG (V)
Supplementary Figure 1. (a) Ids vs. VBG. Inset: optical image of device. (b) Hall voltage VHall as a function of B for chosen backgate bias VBGβs measured at Vds = 1 V. (c) Carrier densities n2D calculated from the slopes of (b) as a function of VBG for Vds = 0.5 and 1 V. (d) Hall mobility for VBG at Vds = 0.5 and 1 V.
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Supplementary Note 2. 2- and 4-probe mobilities
Supplementary Figure 2. (a) Temperature dependent 2-probe (left) and 4-probe (right) mobility for several drain-source voltage Vds. (b) Voltage dependent total (RT) and contact (Rc) resistance at 10K. Resistances were evaluated at backgate bias at which mobilities were calculated.
Supplementary Fig. 2a presents 2- (left) and 4-probe (right) field effect mobilities as a function of T for several Vdsβs, denoted as π2FE and π4FE , respectively. Here, π4FE is replotted for
comparsion. π2FE were calculated from the maximal transconductance (πm β‘ ππΌD βππBG ) within
the range of experimental VBG. In general, π2FE < π4FE due to the contact resistance Rc. The
magnitudes of π2FE and π4FE for Vds show quite different tendency. Focusing on the behavior at
low temperature, π2FE increases with Vds up to 1V, and turns around to decrease as Vds further increases. On the other hand, π4FE keeps decreasing as Vds increases. These contrasting
behaviors originate from Rc and the dependence of Ο on Vds. First of all, Supplementary Fig. 2b
shows the total resistance RT (squares) and the contact resistance Rc (circles) at T = 10K, where
RT = Rc+Rch, Rch is the channel resistance. At low Vds, Rc is dominant due to the Schottky barrier. As Vds increases, Rc becomes smaller and negligible above 1 V because the Schottky barrier becomes thinner. Consequently, π2FE is determined mainly by Rc for low Vds and Rch for high
Vds. Since the maximal transconductance is found in the metallic phase, and Ο for mobility 4
calculation decreases as Vds increases in this metallic phase as shown in Fig. 1b of the manuscript, π2FE shows the non-monotonic behavior for Vds. In contrast, since π4FE involves only the
channel, it monotonically decreases as Vds increases. In Supplementary Fig. 1a, the largest π4FE
at 10K is ~720 cm2V-1s-1 at Vds = 0.1 V. This voltage itself is certainly not small to give the zero voltage limit of conductivity. However, according to Supplementary Fig. 2b, most voltage drop for 0.1 V occurs in the contact, and only ~2% in the channel, which is ~2 mV. Therefore, we believe that the zero voltage limit of π4FE is not far from ~720 cm2V-1s-1, and also the temperature scaling analysis of conductivity obtained at 0.1 V is reasonable.
Supplementary Note 3. Scaling analysis for 3.5 nm-thick MoS2
4.6 ΞΌm ~ 3.5 nm
6.5 ΞΌm 11.1 ΞΌm
Supplementary Figure 3. Optical image of device (left), AFM image (middle), and thickness profile along the black line in AFM image (right). We show the properties of 3.5 nm thick MoS2 sample in this section. Supplementary Fig. 3 shows the optical and AFM images, and thickness profile of this sample. Supplementary Fig. 4 shows conductivities for temperature (4a) and electric field (4d), and scaling analysis. The temperature 4K, where E-scaling is performed, is low enough to be in the diffusive regime, i.e., it is lower than the Dingle temperature πD = βπβ(2ππ΅ πβ π) ~10K for
π~1000 cm2V-1s-1.
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As shown in Supplementary Fig. 5, the conduction in the insulating phase is very well described by Efros-Shklovskii variable range hopping model down to the lowest temperature, 4K. b
a 10-4
c 101
VBG=
102
58
100 10
-7
-2
10
10
101
-3
-8
10
-9
10-4
10
10
zMΞ½M~1.33 0.1
100
1
10
100
T/T0
e
T (K)
101
10-5
100
10-6
10-1
10-8
10-3
10-9
10-4 0
1x105
E (V/m)
2x105
10-5
0.6 0.8
(1+zI)Ξ½I~4.64
10-2
10-7
0.4
|Ξ΄n| 104
Ο/Οc
10-4
0.2
f
E0 (V/m)
10
d
Ο (S)
zIΞ½I~2.06
10-1
T0 (K)
10-6
Ο/Οc
Ο (S)
10-5
103 102
(1+zM)Ξ½M~3.70
101 100 101 102 103 104 105
E/E0
0.2
0.4
0.6 0.8
|Ξ΄n|
Supplementary Figure 4. a ππ vs. T for several VBGβs at Vds = 0.2 V. b Renormalized
conductivity ππβππc by the conductivity at nc as a function of rescaled temperature T/T0. c Temperature scaling parameter T0 vs. |πΏπΏπΏπΏ|. d ππ vs. E for several VBGβs at 4K. e Renormalized
conductivity ππβππc by the conductivity at nc as a function of rescaled electric field E/E0. f
Electric field scaling parameter E0 vs. |πΏπΏπΏπΏ|. Arrows in (a) and (d) indicate the traces at the critical field Vc = 37 V.
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b
a 10-5
-7
10
10-8 10-9
1200 VBG= 34 V 31 V 28 V 25 V 22 V 19 V 16 V 13 V 10 V
0.1
900
TES (K)
Ο (S)
10-6
600 300 0 10 15 20 25 30 35
0.2
0.3 -1/2
T
0.4
0.5
VBG (V)
-1/2
(K )
Supplementary Figure 5. a ππ vs.π β1/2 for Efros-Shklovskii hopping conduction. b Fitting parameters TES.
In the following, we show the simulation result for Joule heating of this device. It is performed using a COMSOL multiphysics modeling software. For the simulation, we used following parameters: β’ sample thickness: 3.5 nm, Au electrode thickness: 60 nm. β’ applied voltage to sample: 2 V β’ sample thermal conductivity1: 60 W/mβK. β’ thermal boundary conductivity between MoS2 and SiO22: 14 MW/m2K. We simulate for three regions at 4K, i.e., insulating, near transition, and metallic phase. For simplicity, they are distinguished by the current level. Since the current level at transition point is ~ 1Γ10-5 A, we choose 8Γ10-6 A for insulating and 7Γ10-5 A for metallic region. According to the results in Supplementary Fig. 6, Joule heating for all three regions in our measurement configuration is insignificant.
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5.02 K
Mean value of βT : 0.09 K
Mean value of βT : 0.12 K
Mean value of βT : 0.84 K
4K
Supplementary Figure 6. Temperature rise (βπ) due to Joule heating at T = 4K. For insulating region (left), βπ~0.09K, near transition (middle), βπ~0.12K, and metallic phase (right), βπ~0.84K. These values are less than 20% independent of the regions.
To demonstrate that the excitation we used for the conductivity measurement in this device is
close to the zero voltage limit so that the heating effect for temperature scaling is insignificant, we show two values in Supplementary Fig. 7 below, one in the zero voltage limit and the other at Vds = 0.1 V. Supplementary Figs. 7a and 7d are the conductivities as a function of channel voltage Vch for several temperatures at VBG = 58 V for metallic and 34 V for insulating phase, respectively. Supplementary Figs. 7b and 7e are the expanded views of 7a and 7d near the zero voltage. The red dotted lines in these figures show the trend of conductivity change for extrapolating the conductivity in the zero voltage limit. Finally, we show two values in Supplementary Figs. 7c and 7f for VBG = 58 V and 34 V, respectively. Closed circles are the values in the zero voltage limit, and open circles are the ones at Vds = 0.1 V which were used for the temperature scaling. There are some underestimation for metallic phase and overestimation for insulating phase at the lowest temperature, but other than that, two values are quite similar. Since the temperature scaling was performed for the broad temperature range, we do not think around 10% error at the lowest temperature does cause the significant errors in the temperature scaling.
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a
b
VBG = 58 V
120 100
c 200 180
200 180
160
160
140
Ο0 (Β΅S)
Ο (Β΅S)
140
4K 6K 8K 10 K 15 K 20 K 30 K 40 K 60 K 80 K 100 K 120 K
Ο (Β΅S)
200 180 160
120 100
0.2
0.4
0.6
0.8
Vch (V)
d 40
4 K, 10 K, 30 K,
6 K, 15 K, 40 K
80
80 0.00
1.0
e 8 K, 20 K,
0.01
0.02
Vch (V)
f
40
Ο0 (Β΅S)
Ο (Β΅S)
Ο (Β΅S)
20 15
0.2
VBG = 34 V
5
0.4
0 0.00
Vch (V)
0.6
T (K) 40
20 10
10
10
20 40 60 80 100 120
30
30 25
0 0.0
0
35
30 20
120 100
80 0.0
140
0 0.01
0.02
0.03
Vch (V)
0
10
20
30
40
T (K)
Supplementary Figure 7. a and c Voltage dependent conductivities at VBG = 58 V and 34 V for several temperatures, respectively. b and e Expanded views of (a) and (d) near 0 V, respectively. Red dotted lines indicate converging trends to the conductivity in the zero voltage limit. c and f Solid circles are the conductivity values in the zero field limit and open circles are the values at Vds = 0.1 V. Vch corresponding to this Vds value is β² 3 mV.
Supplementary Note 4. Intermediate glass phase Electron glass features were experimentally observed in strongly disordered Si-MOSFET3. In the ref. S3, the critical carrier density πΏπΏc β 5.2 Γ 1011 cm-2. πΏπΏg β 7.5 Γ 1011 cm-2 is identified as a carrier density below which the 2D electron system freezes into an electron glass. πΏπΏsβ is
determined such that πππβππ = 0 at this carrier density. The data close to nc are well described
by the power law behavior Ο(πΏπΏ2D , π) = π(πΏπΏ2D ) + π(πΏπΏ2D )π π₯ with π₯ β 1.5. This feature is consistent with the theoretical prediction of the existence of an intermediate glass phase in
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πΏπΏc < πΏπΏs < πΏπΏg < πΏπΏsβ (ref. 9 in the manuscript). In our data, we identified VBG = 10 V as a critical field corresponding to πΏπΏc β 3.37 Γ 1012 cm-2. The real critical field could be in between 10 and
15 V as shown in Supplementary Fig. 8 below. If it really is, the temperature dependence of Ο
would be weaker than the one at VBG = 10 V . The power x for the trace at VBG = 10 V is 0.91 far from 1.5, and the intermediate region (colored), if there is, very narrow, i.e., Ξ΄πΏπΏg β‘ οΏ½πΏπΏg β πΏπΏc οΏ½βπΏπΏc β 0.11, which is contrast to the Si case in ref. S3, Ξ΄πΏπΏg β 0.44. In this sense, an apparent metallic glass feature is not visible or exists in a very narrow range.
6x10-5 4x10-5
Ο (S)
x=1.5
x=0.91 15 V 10 V
2x10-5
10
100
T (K) Supplementary Figure 8. Ο vs. T for two traces at VBG=10 and 15 V just near the MIT (5 nm thick MoS2).
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Supplementary Note 5. Scaling analysis for monolayer MoS2
Supplementary Figure 9. a Backgate VBG dependent conductivity in the unit of e2/h at Vds = 0.5 V for various temperatures. b Temperature dependent 4-probe mobility for VBG = 50 V. c Conductance G as a function of temperature for various VBGβs. d Renormalized conductivity ππβππc as a function of temperature for various VBGβs. e Renormalized conductivity for rescaled
temperature T/T0. f Temperature scaling parameter T0 vs. |πΏπΏπΏπΏ|.
Temperature scaling was performed for monolayer MoS2. CVD (chemical vapor deposition) grown monolayer MoS2 is transferred onto the h-BN film (~20 nm) mechanically exfoliated on the SiO2 (300 nm)/Si substrate. The optical image of device with metal (Cr/Au) electrodes is shown in the inset of Supplementary Fig. 9a. Supplementary Fig. 9a presents the backgate bias dependent conductivity in the unit of e2/h for selected temperatures. This conductivity was taken at Vds = 0.5 V which is not small but we believe that this does not change the scaling behavior significantly since our temperature range for scaling is rather high, T > 60K so that the field effect is relatively weak compared to the thermal effect. Conductivity crossover for temperature 11
occurs around VBG ~ 23 V signifying the metal-insulator transition. The critical carrier density nc at this bias is estimated to be ~1.9 Γ 1012 cm-2 using πΏπΏ2D = πΆox (πBG β πth )/π at room temperature as in the manuscript. This nc yields rs~7.8.
Temperature dependent 4-probe mobility is calculated at VBG = 50 V and shown in Supplementary Fig. 9b. The mobility at ~10K is approximately 170 cm2V-1s-1. Compared with multilayer, monolayer MoS2 is less interacting and more disordered system. Supplementary Figs. 9c and 9d display the conductance and renormalized conductivities by the critical conductivity as a function of temperature, respectively. Supplementary Fig. 9e shows the collapse of renormalized conductivities after rescaling the temperatures. Finally, the critical exponent π§π~2.87 is obtained from the linear fit of temperature scaling
parameter T0 for |πΏπΏπΏπΏ| as shown in Supplementary Fig. 9f. In addition to this large value, more
symmetric scaling parameter T0 for the metal and insulating phase suggest that MIT in this monolayer MoS2 is likely disorder driven. The deviation at higher |πΏπΏπΏπΏ| in metallic regime is not
clearly understood. The intermediate state may exist.
In another CVD-grown monolayer MoS2 on h-BN, we measured voltage-dependent conductivity for several temperatures to see how much variation of conductivity for the voltage changes as the temperature increases. Supplementary Fig. 10a shows the optical image of monolayer MoS2 on h-BN. Supplementary Fig. 10b displays the backgate bias-dependent conductivity for several different temperatures taken at Vds = 0.5 V. The metal-insulator crossover is not visible for πBG β€ 80 V but the larger curvature for smaller temperature
indicates MIT to occur at higher VBG. Supplementary Figs. 10c and 10d show drain-source voltage-dependent conductivity for different temperatures at VBG = 35 V and 70 V, respectively.
The conductivity changes with Vds. The changing rate is stronger at lower temperature, while it becomes weaker as temperature increases. For T > 100K and Vds 60K as noted earlier and the collapse (Supplementary Fig. 10e) is quite good up to 280K, this suggests that the scaling analysis with this voltage Vds = 0.5 V is still reliable for insulating phase. Accordingly, for metallic phase, we expect the scaling analysis for monolayer to be still valid, although we could not explicitly proved it due to the inaccessibility of metallic phase in 12
additional experiments. Although the data for monolayer is not fully comprehensive, we believe it is worth reporting, since, to our knowledge, this is the first report for any kind of atomic monolayer form of materials.
Supplementary Figure 10. a Optical image of monolayer MoS2 on h-BN. b Backgate bias dependent conductivity for several temperatures taken at Vds = 0.5 V. c Drain-source volate Vds dependent conductivity for several temperatures at VBG = 35 V and d 70 V.
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Supplementary Figure 11. AFM image of multilayer MoS2 device (left) and thickness profile along the line.
Supplementary References 1.
Bae J. J., et al. Thickness-dependent in-plane thermal conductivity of suspended MoS2 grown by chemical vapor deposition. Nanoscale 9, 2541-2547 (2017).
2.
Yalon E., et al. Energy Dissipation in Monolayer MoS2 Electronics. Nano Lett. 17, 34293433 (2017).
3.
Bogdanovich S., Popovic D. Onset of glassy dynamics in a two-dimensional electron system in silicon. Phys. Rev. Lett. 88, 236401 (2002).
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